The curve complex is a complex and is invariant under normal-form moves

The complex and shift. The square-zero verification is [L1], already proved for the actual essential-segment/path rules of the current definition. A deck shift changes each crossing index by $(r_1,r_2)$, so every summand changes by $[-r_1]\{r_2\}$. The source keeps its path entries, while the homological shift changes their sign by $(-1{)}^{r_1}$; the supplier’s sign map $(-1{)}^{r_1{x}_1}$ on the summand indexed by $x$ gives the stated chain isomorphism.

The crossing correspondence. Use [L2] to transport $\tilde{c}$ to ${\tilde{c}}^{\mathrm{\prime }}$, preserving the dividing curves setwise. A crossing with $d_i$ moves along $d_i$ and retains its index $i$, its two local indices by [L3], and the types and orientations of its incident essential segments. Thus its source and target summands are the identical shifted projective. Define $\mathrm{\Phi }$ on that summand to be the identity into the summand indexed by the transported crossing; define $\mathrm{\Psi }$ by the inverse correspondence. Every differential entry is multiplication by the same path before and after transport, so $\mathrm{\Phi }d=d^{\mathrm{\prime }}\mathrm{\Phi }$, $\mathrm{\Psi }{d}^{\mathrm{\prime }}=d\mathrm{\Psi }$, and $\mathrm{\Psi }\mathrm{\Phi }=\mathrm{i}\mathrm{d}$, $\mathrm{\Phi }\mathrm{\Psi }=\mathrm{i}\mathrm{d}$. No arbitrary overall deck shift is introduced for isotopic bigraded representatives.

Explicit cone contraction. In the degree-increasing convention the mapping cone of $\mathrm{\Phi }$ is $L({\tilde{c}}^{\mathrm{\prime }}{)}^n\oplus L(\tilde{c}{)}^{n+1}$ with differential $D(y,x)=(d^{\mathrm{\prime }}y+\mathrm{\Phi }x,-dx)$. Define $h(y,x)=(0,\mathrm{\Psi }y)$. Then $Dh+hD=(y,-d\mathrm{\Psi }y+\mathrm{\Psi }{d}^{\mathrm{\prime }}y+x)=(y,x)$ by the chain-map identity. Hence the cone is contractible, and in particular the two complexes are isomorphic in $C_m$; the displayed maps are stronger than merely homotopy inverses.

Presentations and conclusion. Stretching the curve drawing, grouping the indexed summands into columns, and folding arrows into a complex leave precisely the same summands and differential entries, so their comparisons are the corresponding reindexing chain isomorphisms of step 1.2. Inessential end segments carry no pair of crossing modules and are omitted in the construction, not cancelled through a fabricated identity pivot. Half twists have their separate generator action. The isotopy isomorphism and deck-shift rule therefore prove the full claimed invariance and compatibility without additional choice.